THE MODAL AND FLOW VELOCITY CORRECTIONS OF MICROPHONE TURBULENCE SCREENS J. L. DAVY

Transcription

1 THE MODAL AND FLOW VELOCITY CORRECTIONS OF MICROPHONE TURBULENCE SCREENS J. L. DAVY RMIT University, School of Applied Sciences, GPO Box 476V, Melbourne, Victoria 31, Australia and Commonwealth Scientific and Industrial Research Organisation, Manufacturing and Materials Technology, PO Box 56, Highett, Victoria 319, Australia Citation: Davy, J 7, 'The modal and flow velocity corrections of microphone turbulence screens', Journal of Sound and Vibration, vol. 36, pp Corresponding Author A/Prof J. L. Davy RMIT University, School of Applied Sciences, GPO Box 476V, Melbourne, Victoria 31, Australia Facsimile Telephone john.davy@rmit.edu.au

2 Abstract Microphone turbulence screens are used to suppress turbulent pressure fluctuations when measuring the acoustic pressure inside a duct with flow. They consist of a long tube with a slit covered with porous material, and thus are also called sampling tubes. Because they are not omnidirectional, it is necessary to calculate corrections when higher order modes are propagating in the duct. In order to calculate these corrections it is necessary to know the directivity of the microphone turbulence screen, the propagation direction and energy of the duct modes and the flow velocity of the air in the duct. This paper derives a theoretical formula for the directivity of a microphone turbulence screen. It shows that this theoretical formula agrees better with experimental directivity data than the previously used empirical directivity formula. Because the empirical directivity formula is not a function of the flow velocity, previous research has separated the modal correction from the flow velocity correction. It is shown that it is not theoretically valid to separate the corrections, and that doing so can lead to large errors at high frequencies in the outlet duct. A new method of calculating the modal correction with flow is presented. This method uses a statistical room acoustics approach in contrast to the deterministic numerical approach of the older method. The new method requires much less computing. It is shown that the new method agrees fairly well with the old method for modal corrections without flow. The new method is compared with experimental measurements of the combined modal and flow velocity corrections. Although the trend is the same, the experimental results are higher than the theoretical results in the mid frequency range. The new method agrees reasonably well with the corrections given in ISO 5136:3. Keywords: Microphone, turbulence, screen, modal, correction, duct

3 3 LIST OF SYMBOLS a y,a z a a 1,a A b c C d d y,d z D e E f f(θ) F g internal dimensions of rectangular cross section duct specific acoustic admittance ratio boundary condition constants microphone turbulence screen internal cross sectional area microphone turbulence screen slit width speed of sound in air modal and velocity correction factor distance from duct wall distance of measurement rectangle from walls of duct diameter of cylindrical duct exp(1) complex amplitude of external sound pressure wave frequency desired ideal pressure squared response effective length ratio of microphone turbulence screen 1 + ( β / k ) G -ik β H i i int() j J 1 () k increase in average pressure squared value due to reflection at wall square root of minus one unit vector in direction of positive x axis function that produces the integer part of a number integer Bessel function of the first kind of order one wave number amplitude k ω /c k 1 k c k j real part of wave number inside screen wave number at third octave band centre frequency values of wave number used to calculate average

4 4 k p magnitude of wave number projection onto duct cross section k x,k y,k z components of wave number in x, y and z directions k k mn K K' L m m 1 M n N wave number vector projection of wave number of duct mode onto duct cross section empirical directivity constant empirical directivity constant length of microphone turbulence screen slit complex wave number inside screen or duct mode index complex wave number inside extension tubes Mach number duct mode index number of points used to calculate third octave band average N(k p ) number of duct modes with wave number projection less than k p p p i,p r p x p complex sound pressure amplitude incident and reflected sound pressure waves complex external sound pressure amplitude at slit complex amplitude of external sound wave p(t,x) external sound pressure at time and position x P sound pressure q complex amplitude of rate of volume addition per unit volume q,q+ q limits of small range of ky values Q r R S t v v w(θ) rate of volume addition per unit volume specific airflow resistance relative measurement radius cross sectional area of duct time velocity of external moving media velocity vector of external moving media angular distribution of sound energy in duct

5 5 x x x x,y,z Z s β β 1 β 1 θ ρ ω ω / n spatial position vector relative to external moving media spatial position on x axis relative to stationary coordinates spatial position vector relative to stationary coordinates spatial variables specific acoustic impedance of surface ρ cb/(ra) value of β in extension tubes attenuation of sound pressure per unit distance inside screen angle of incidence of sound ambient density of air angular frequency when moving with external media angular frequency at stationary point Laplacian operator normal gradient into surface <> average value

6 6 1. INTRODUCTION Microphone turbulence screens (see Fig. 1) are used to suppress turbulent pressure fluctuations when measuring the acoustic pressure inside a duct with flow. They consist of a long tube which is typically about 4 mm long and 13 mm in cross section. Normally there is a 13 mm diameter microphone at one end and a reflecting surface at the other end which is often fitted with an external nose cone. The screen has a narrow (~1 mm) slit which runs the full length of the tube and is covered with porous material, or is manufactured from a porous material. They are also called sampling tubes. The research work described in this paper arose out of a research project to develop a flush mounted microphone turbulence screen for use in a power station chimney flue. The development project required the determination of the sound pressure at the inside wall of the chimney flue. Thus the modal and flow velocity corrections in the International Standard ISO 5136 [1, ] could not be used. The corrections in ISO 5136 are for estimating the sound power propagating along a duct and thus assume a desired microphone sound pressure squared response proportional to the cosine of the angle of incidence relative to the axis of the duct. For the development project, the desired microphone response was omnidirectional. Hence the modal corrections would have larger magnitude. Thus new modal and velocity corrections had to be calculated. A new method of calculating these corrections was developed and it was shown that the modal and velocity corrections could not be separated as is done in ISO 5136:199 [1]. The new method is presented in full in this paper. Since the development of the method described in this paper, ISO 5136:3 [] has been published. The new version of the standard contains combined modal and flow velocity corrections which were calculated using a significant improvement of the method used to calculate the modal corrections in the earlier version. The improved method is different from the method described in this paper.

7 7 The microphone turbulence screen which was developed used anechoic terminations in conjunction with a flush mounted microphone. This is in contrast to the conventional microphone turbulence screen which uses reflecting ends, one of which is the microphone diaphragm. The new method of calculating the modal and velocity correction was applied to a conventional microphone turbulence screen and used to calculate the corrections for estimating the sound power flowing down a duct. In order to do this, a Waterhouse style correction taking account of the variation of sound pressure squared across the duct had to be developed. These new corrections were compared to existing theoretical and experimental estimates of the corrections. It was shown that the differences between the theoretical corrections were mainly due to different assumptions about the directivity of the microphone turbulence screen, and the angular distribution of the sound energy in a duct. The theoretical derivation of the microphone turbulence screen directivity was made more exact, and the theoretical directivity formula was shown to agree better with experimental measurements than the previous empirical formulae.. THEORETICAL DIRECTIVITY The theoretical response of a microphone turbulence screen has been analysed by Neise [3], Wang and Crocker [4] and Michalke [5, 6]. The analysis is extended in this paper to cover the case of a microphone turbulence screen with long tubes of the same crosssectional area at each end of the slit (see Davy and Dunn [7, 8]). A microphone turbulence screen is shown in Fig. 1. The directional response of a microphone turbulence screen can be determined theoretically by studying the propagation of sound within the microphone turbulence screen for different external excitations. The wave equation for sound pressure is 1 p Q - -, P c t = ρ t (1)

8 8 where P is the sound pressure, c is the speed of sound, t is time, ρ is the ambient density, and Q is the net rate of volume addition per unit volume. If the variation of Q with time is sinusoidal, then P will also be sinusoidal in the steady state. Thus Q q P p iωt iωt = e and = e, () where ωo is the angular frequency. The wave equation becomes p+ k p = -i ω ρ q, (3) where k = ω /c is the wave number. For a microphone turbulence screen with an internal cross sectional area A of its tube, a slit width b and a porous fabric covering the slit of specific airflow resistance r, the rate of volume addition per unit volume is b( px p) q =, (4) ra where p x is the complex external sound pressure amplitude. The specific airflow resistance of the fabric is the ratio of the pressure difference across the fabric to the linear velocity of airflow just outside the surface of the fabric. The one dimensional wave equation for a microphone turbulence screen is p x + k p = -i kβ( px - p), (5) where ρcb β =, (6) ra and x is the spatial variable. This equation can be rearranged into p x + mp= Gp, x (7) where

9 9 m k kβ = -i, (8) and G = -i k β. (9) Taking the square root of Eq. (8) with positive real part gives m= k -i β. (1) 1 1 Define β g = + k 1. (11) Then g + 1 k1 = k, (1) and g 1 β1 = k. (13) If the external sound pressure is sinusoidal in space px = E -i e kx x, (14) and the wave equation becomes p x -ikx x + mp= GEe. (15) The solution of this equation is ikx x GE e p= + a e + a e m k x -imx imx 1. (16)

10 1 This solution is only valid if β is independent of x. This will always be assumed to be the case in this paper. The first term of the solution is a particular integral of Eq. (15) and the last two terms are the solution of the homogeneous version of Eq. (15). The constants a 1 and a have to be determined from the boundary conditions. The boundary conditions for a sound wave are P ρ P + =, n Z t where Zs is the specific acoustic impedance of the surface and s P n (17) denotes the gradient of P in the direction normal to the surface and from the space into the surface. The specific acoustic impedance of a surface is the ratio of the sound pressure at the surface to the particle velocity in the direction normal and into the surface. For a sound pressure which varies sinusoidally with time Eq. (17) can be rewritten as p + ikap =, n (18) where the specific acoustic admittance ratio a is given by a ρ c Z =. (19) s For a rigid surface a is equal to zero. If the tube continues on to plus or minus infinity, past the ends of the microphone turbulence screen slit that is exposed to the external sound, the sound pressure is given by the second or third term of the right hand side of Eq. (16) providing an appropriate value β 1 of β is used. The first term is zero because there is no external sound pressure acting on the tube extension and a 1 or a is zero since there cannot be any sound coming from infinity. At the end of the tube which extends to plus infinity p d p = = -imx 1 -ima 1 1e = i mp 1, n d x ()

11 11 where m 1 is the value of m in the extension tube. Thus a 1 = m/k. If the tube extension has no slit then β 1 =, m 1 = k and a = 1. If the slit and its covering fabric in the extension tube are the same as the slit and covering fabric exposed to the external sound in the actual turbulence screen, an anechoic termination is obtained. In this case β 1 equals β, and the m 1 for the extension tube is the same as the m for the actual turbulence screen. If the actual microphone turbulence slit runs from x = to x = L and the specific acoustic admittance ratio is the same at both ends of the slit, the boundary conditions are d p () = i kap (), (1) d x d p ( L) = -i kapl ( ). () d x A large amount of tedious algebra enables Eqs. (16), (1) and () to be solved for the constants a 1 and a, which can then be substituted back into Eq. (16) to obtain the sound pressure at x = L. This gives i i i ( ) i ( ) e k x L ml ml pl k β k a+ kx m + ( ka kx)[( m+ ka)e + ( m ka)e ] = 1. iml iml (3) px( L) m kx ( m+ ka) e ( m ka) e If a =, Eq. (3) becomes iml i( kx m) L pl ( ) ikβ kx 1+ e e = 1. + iml px( L) m kx m 1 e (4) This equation agrees with Eq. (5) of Michalke [5] as modified by the errata. It applies for rigid reflecting surfaces at both ends of the slit. If a = m/k, Eq. (3) becomes i( kx m) L ( ) pl ( ) ikβ 1 e = - p ( L) m( m k ) x x (5) This applies for the case of an anechoic termination at both ends of the slit.

12 1 The complex arithmetic can be removed from Eq. (5) by taking the modulus squared of both sides and substituting for m using Eq. (1). The result is pl ( ) k β 1+ e e cos[( k kx ) L] =. p ( L) k ( k k ) x β1l β1l β1 1 x + β1 (6) This equation agrees with Eq. (35) of Neise [3] if Neise's approximations are corrected and his pressure doubling factor is removed. Neise's approximations are k 1 = k, β 1 = β o and β 1 «k 1. The moduli squared of Eqs. (3) and (4) were evaluated by programming computer spreadsheet functions to perform complex arithmetic. The equation of a plane sound wave is ptx i( ωt- kx. ) (, ) = p e, (7) where p is the sound pressure at time t and position x relative to the medium, p o is the complex amplitude of the sound wave, ω is the angular frequency of the sound wave and k is the wave number vector. If the medium is moving at velocity v, then relative to the medium, a stationary point appears to move with a velocity -v. Its position relative to the medium is given by x -vt where x is its position relative to stationary coordinates and where the moving medium coordinates correspond with the stationary coordinates at time t =. The sound pressure at this stationary point is given by ptx i[( ω+ kv. ) t - kx. (, ] ) = pe. (8) If v is in the direction of the positive x axis and x is on the x axis, then v = vi and x o = x i. The sound pressure is given by ptx i[( ω+ kvtkx x ) - x (, ] ) = pe, (9) where k x is the component of the wave number k in the direction of the x axis. If the sound wave is travelling at an angle of θ to the x axis then kx ω cosθ = kcos θ =, (3) c

13 13 where k is the amplitude of k and c is the speed of sound. If the Mach number is M = v/c, then ptx i[(1 + M cos θ ) ωt - kx (, cos θ ] ) pe. = (31) Thus the angular frequency of the sound wave observed at the stationary point is ω = (1+ M cos θω ). (3) If a stationary microphone turbulence screen is placed along the x axis in the moving medium and excited by the plane sound wave in the moving medium, the wave number k inside the turbulence screen is given by k ω c (1+ M cos θ ) ω c = = = (1+ M cos θ ) k, (33) and from Eqs. (3) and (33) k x k cos θ =. (34) 1 + M cosθ The theoretical response and directivity of a microphone turbulence screen can be calculated using Eqs. (3) and (34). The theoretical response and directivity will not be valid when the cross sectional dimensions of the turbulence screen become comparable to the wavelength. The response will be inaccurate when the cross sectional dimensions become equal to a quarter of the wavelength because Eq. (4) fails. This is because the external sound pressure acting through the resistance of the material covering the slit is no longer driving an acoustic volume compliance but the zero acoustic impedance of a quarter wavelength depth. The directivity will be incorrect when the cross sectional dimensions become equal to half of the wavelength because Eq. (3) will no longer be valid. This is because cross mode propagation becomes possible and the one dimensional wave equation is no longer adequate. For the microphone turbulence screens considered in this paper with a typical cross sectional dimension of 13 mm, this cross sectional

14 14 dimension becomes equal to a quarter of the wavelength at 6.6 khz and half of the wavelength at 13. khz. 3. MODAL AND VELOCITY CORRECTION A microphone turbulence screen is conventionally calibrated in an anechoic room with zero flow ( Mach number M equals zero ) and with angle of incidence equal to zero. For non-zero Mach numbers and non-zero angles of incidence a theoretical correction to the calibration must be calculated. If sound is incident from different directions at the same time, the theoretical correction must be averaged over the different angles of incidence with a weighting which is proportional to the sound energy incident from each direction. This approach assumes that sound incident from different directions is uncorrelated. If the duct is anechoically terminated or has an open end whose dimensions are larger than a wavelength, back reflections can be ignored. This means that it is only necessary to average over angles of incidence from to 9. The angular distribution of sound energy in a duct is not normally known. The obvious assumptions that might be made about the angular distribution of sound energy in a duct are that every mode carries equal power down the duct, that every mode has equal energy density, that equal energy is incident from every element of angle of incidence or that equal energy is incident from every element of solid angle. The correction factor C(ω,M) is calculated by averaging the pressure-squared response p ( ωθ,, M) ( ω,,) p of the microphone turbulence screen with the appropriate sound energy angular distribution weighting factor w(θ) over angles of incidence θ from to 9 and dividing this into the average of the desired angular response with the weighting function. This gives C ( ω M) ( ω ) ( ω ) π / f() + f( θ) w( θ)dθ, =. π / ( ω θ ) ( ω ) p,, M p,, M + w( ) d p,, θ θ p,, (35)

15 15 The first terms in the numerator and denominator take account of the plane wave mode. For frequencies below the cut on frequency of the first duct cross mode, the second terms in the numerator and denominator are set equal to zero, because there are no propagating cross modes. Using Eqs. (35), (3) and (34), C(ω,M) can be calculated theoretically. The function f(θ) is the desired ideal pressure squared response as a function of angle of incidence. For measurements of sound pressure squared f(θ) is equal to 1. For measurements of sound power propagating down a duct, f(θ) is equal to cosθ, since the sound power is proportional to the projection of the duct cross sectional area onto a plane perpendicular to the direction of propagation of the sound. This projected area is proportional to cosθ. For equal energy from every angle of incidence, w(θ) is constant. For equal energy from every element of solid angle, w(θ) is proportional to sin(θ). For every mode with equal energy density, w(θ) is proportional to the number of modes per unit angle of incidence. If every mode carries equal power down the duct, then the modal energy densities are proportional to 1/cosθ since the power carried down the duct is proportional to cosθ. In this case, w(θ) is proportional to the number of modes per unit angle of incidence divided by the cosine of the angle of incidence. The correction factor C is the factor by which the desired values are greater than the values measured with the microphone turbulence screen. Thus the values measured with the microphone turbulence screen must be multiplied by the correction factor. In practice the correction factor will be expressed in decibels and will be added to the measured sound pressure level. Since the correction factor will usually be positive, applying it will normally mean increasing the measured sound pressure level. The formula for the number of modes per unit angle of incidence will be derived for a rectangular cross section duct, since it is well known that the formula for the number of cross modes depends asymptotically only on the area of the cross section and the wave

16 16 number. (See for instance Balian and Bloch [9] and note that area is the two dimensional equivalent of three dimensional volume). If the size of a rigid walled rectangular duct cross section is a y by a z, the modal wave number vectors k mn of the cross modes are (mπ/a y, nπ/a z ), where m and n are any nonnegative integers. The k mn form a regular lattice in the first quadrant of the k y k z plane with each point occupying an area in this two dimensional k space of π /S, where S = a y a z is the cross sectional area of the duct. The area of the first quadrant containing wave number vectors k whose magnitude is less than k p is πk p /4. Thus the number of modal wave number vectors less than k p is Sk p /(4π). The modal wave number vectors k m and k n which lie on the k y and k z axes have only been half counted since half their area lies in other quadrants. The total number of modal wave number vectors on an axis is (a y +a z )k p /π, and counting an extra half for each of these vectors gives the total number N of modal wave number vectors less than k p, for the square case a y equals a z, as ks p + 4kp S Nk ( p ) =. (36) 4π The number of modes per unit of wave number magnitude is d N ks p + 4 S =. (37) dk 4π p For a mode propagating down the duct, the square of the magnitude of its modal wave number is k = k + k (38) x p, where k p is the magnitude of the projection of its modal wave number vector onto the cross sectional area of the duct. Now k p = ksinθ where θ is the angle between the direction of propagation of the mode and the centre line of the duct. Thus dk p /dθ = kcosθ. Hence the number of modes per radian is

17 17 dn d k p k Ssinθ + 4k S cosθ w( θ ) = =. (39) dk dθ 4π p If every mode carries equal power down the duct, the weighting function is obtained by dividing the number of modes per radian by the cosine of the angle of incidence. This gives kssinθ+ 4k S w( θ ) =. (4) 4π In actual calculations with Eqs. (39) and (4), k will be approximated with k. Because the modulus squared of Eq. (3) cannot be integrated analytically, the integral in the denominator of Eq. (35) must be integrated numerically. In this report the trapezoidal rule was used with steps of 5. However for the choices of f(θ) and w(θ) given above, the integral in the numerator of Eq. (35) can be integrated analytically. 1. For w(θ) = 1 and f(θ) = 1, π / π f( θ) w( θ)d θ =. (41). For w(θ) = 1 and f(θ) = cosθ, π / f( θ) w( θ)dθ = 1. (4) 3. For w( θ ) = ( k Ssinθ + 4k S cos θ)/(4 π ) and f(θ) = 1, π / ks+ 4k S f( θ) w( θ)d θ =. (43) 4π 4. For w( θ ) = ( k Ssinθ + 4k S cos θ)/(4 π ) and f(θ) = cosθ, π / ks+ 3π k S f( θ) w( θ)d θ =. (44) 1π 5. For w(θ) = sin(θ) and f(θ) = 1,

18 18 π / f( θ) w( θ)dθ = 1. (45) 6. For w(θ) = sin(θ) and f(θ) = cosθ, π / 1 f( θ) w( θ)d θ =. (46) 7. For w( θ ) = (k Ssinθ + 4 k S)/(4 π ) and f(θ) = 1, π / ks+ π k S f( θ) w( θ)d θ =. (47) 4π 8. For w( θ ) = (k Ssinθ + 4 k S)/(4 π ) and f(θ) = cosθ, π / ks+ 4k S f( θ) w( θ)d θ =. (48) 4π If the weighting function is relatively constant, the integrals in Eq. (35) are basically measures of angular bandwidth. This means that directivity values which are more than 3 db down will have little effect on the correction factor C. Thus for determining the correction factor C it does not matter greatly if our predictions or measurements of directivity are in error for those values which are more than 3 db down. The modulus squared of Eq. (3) has to be evaluated for third octave bands of noise. For those frequencies where the rate of change of Eq. (3) as a function of frequency across the third octave band is relatively constant it is sufficient to evaluate the modulus squared of Eq. (3) at the centre frequency of the third octave band. Equation (3) contains oscillating exponential functions with arguments of the form ik x L or ik 1 L. For low Mach numbers, k x and k 1 are approximately equal to or less than k. By the sampling theorem, a new evaluation point should be used at least every time k L increases by π. Thus the number of points used to calculate a third octave band average is given by N = + (49) 1/6-1/6 1 int[( - ) kl/ π ],

19 19 where int() is the function that produces the integer part of a number. The average of the modulus squared of Eq. (3) is averaged over the N values of k j given by k k c (j-n-1)/(6 N) j =, (5) where k c is the value of k at the centre frequency of the third octave band. Since the microphone turbulence screen integrates in the sound pressure domain, the effective length of the microphone turbulence screen slit can be estimated by - 1 integrating e β x over the length of the slit from x equals zero to x equals L. If the effective length is divided by the length L of the slit it gives the effective length ratio F = -β1l (1-e )/( β1l). (51) This gives the fraction of the slit length over which the microphone turbulence screen appears to effectively sample. 4. WATERHOUSE CORRECTION If we only wish to measure the sound pressure squared, or the sound intensity in the direction parallel to the centre line of the duct, at the position of the microphone turbulence screen in the duct, then the modal and velocity corrections derived in the previous section are all that need be applied. However we usually wish to estimate the sound pressure squared or the sound intensity averaged across the entire cross sectional area of the duct. In this situation, it is necessary to apply a Waterhouse [1] correction to account for the fact that the sound pressure is greater near the walls of the duct, because of the increase in sound pressure that occurs when sound is reflected at a rigid surface. In a duct, unlike in a reverberation room, the microphone will often be in the interference pattern created near the duct wall by the reflections. This means that the distance of the microphone from the duct wall must also be taken into account. If we have a plane wave of unit amplitude incident upon a rigid surface in the x z plane from the positive y half space, its sound pressure will be given by the equation

20 i( ω tkx - x + kyykz - z ) pi = e, (5) and the reflected sound pressure wave will be p r i( ωtkxk - x - yykz - z ) = e. (53) At time t =, the sound pressure on the positive y axis ( x =, y >, z = ) will be the sum of p i and p r. Thus the sound pressure is ikyy -ikyy p = e + e = cos( ky y ), (54) and the modulus squared of the sound pressure is p = 4cos ( kyy) = [cos( kyy) + 1]. (55) The average value of the modulus of the sound pressure squared is p =, (56) and normalising Eq. (55) by dividing by Eq. (56), gives p p = 1+ cos( ky). (57) y Propagating rectangular duct modes with wave number k have values of k y and k z which lie uniformly spread in the quarter circular quadrant in the k y k z plane bounded by k k k k y, z, y + z k. (58) The area of this quarter circle quadrant is πk /4. The area of this quadrant with values of k y between q and q+ Δ q is k q q Δ. Thus if all propagating modes have the same amplitude and are uncorrelated, the average value of Eq. (57) over all propagating modes is k 4 H = k ky[1+ cos( kyy)]d. π k k y (59)

21 1 Performing the integration yields H J( ky) ky 1 = 1 +, (6) where J1() is the Bessel function of the first kind of order one. The average increase in pressure squared across the duct due to the reflection at one duct wall is H ay ay 1 ky 1 ky (1 ) d y 1 d. 1 J ( ) 1 J ( ) = + = + a ky a ky y (61) y y For values of k for which ka y is less than π, cross modes with a non zero wave number component in the direction of the y axis cannot exist. In this situation H is equal to one. The integrand of the last integral in Eq. (61) is an oscillating function which decays rapidly as ky increases. For values of k for which ka y is greater than or equal to π, this integral can be approximated by replacing the upper limit of the integral with plus infinity. This gives H J1 ky 1 dy 1 1 ( ) 1 = + a = +. (6) ky ka y y Taking account of the other three duct walls and ignoring interactions at the four corners gives H = 1 +, ka + ka (63) y z where the second and/or third terms are set equal to zero if ka y and/or ka z are less than π.. The second term of Eq. (6) is an oscillating function which decays rapidly with increasing distance. If the sound pressure squared is measured at a distance d from the duct wall such that kd is much greater than π, then H is approximately equal to one, and the measured sound pressure squared must be multiplied by Eq. (63) to obtain the average sound pressure squared across the duct. In practice the microphone will often be closer to the duct walls than π/k and the measured pressure squared has to be

22 corrected for the position of the microphone using Eq. (6). If the microphone is moved over a rectangle whose sides are at a distance of d y and d z from the walls of the duct in the direction of the y and z axes, the dimensions of the measurement rectangle are a y - d y and a z - d z. Using Eq. (6) for each side of the measurement rectangle and the side of the duct nearest to it, and averaging with a weighting equal to the length of the side of the measurement rectangle, gives H ( a d )J ( kd ) ( a d )J ( kd ) = + + ( a + a d d ) kd ( a + a d d ) kd z z 1 y y y 1 z 1. y z y z y y z y z z (64) If the ratio of the width of the measurement rectangle to the width of the duct is the same in both the y and z axes directions, the ratio will be called the relative radius and denoted by R. In this case Eq. (64) can be written as H a J ( ka [1 R]) a J ( ka [1 R]) = + + ( a + a ) ka (1 R) ( a + a ) ka (1 R) z 1 y y 1 z 1. y z y y z z (65) If ka y and/or ka z are less than π, the corresponding J 1 () function is set to zero since there are no cross modes with wave number components in that direction. Because the J 1 () function oscillates, when calculating third octave band average values, it is set to zero for values of its argument greater than its second positive zero. Its second positive zero is 7.. Thus to estimate the average sound pressure squared across the cross sectional area of the duct, it is necessary to divide the measured sound pressure squared by Eq. (65) and multiply it by Eq. (63). Values for circular ducts are calculated using a square cross sectioned duct of the same area. According to Morse and Ingard [11], the cut on frequency of the first cross mode in a cylindrical duct is.5861 c / D where D is the diameter of the duct. The cut on frequency of the first cross mode for a square cross sectioned duct with the same cross sectional area as the cylindrical duct is c/( π D) =.564 c/ D. This differs by less than 4% from the true cylindrical value. Thus approximating a cylindrical with a square cross sectioned duct is unlikely to lead to large errors.

23 3 5. EXPERIMENTAL DIRECTIVITY Research work on the directivity of microphone turbulence screens has been dominated by an empirical expression developed at Purdue University. The expression is [13, 14, 1, 1, ] p( θ ) 1 1 = =, p + klθ K + fθ K 3 3 () 1 1 ' (66) where the empirical constants K and K' are related by the equation π LK K ' =. (67) c The values of the constant will be given for the case where the angle of incidence θ is expressed in radians. According to Bolleter [1], Flory and Crocker [13] originally developed the second version of Eq. (66) with a value of K' equals.35 for frequencies below khz. Bolleter then states that it was later found that K' equals.45 gave a better approximation. The first form of Eq. (66) was given by Bolleter, Cohen and Wang [14] with K equals.61 below khz. K equals.61 corresponds to Bolleter's value of K' equals.45 for a temperature of C and a slit length of 4 mm. It should be noted that none of these researchers claimed that the empirical formula could be used above khz. The ISO standard 5136 [1, ] gives upper and lower limits for directivity. The upper limit uses Flory and Crocker's value of K' equals.35 for frequencies of 1,, 4 and 8 khz. The lower limit uses K' equals.15 for 1 and khz, and K' equals. for 4 and 8 khz. Neise, Frommhold, Mechel and Holste [15] used the value of K' equals.5. The values of the empirical constants K and K' for a temperature of C and a slit length of 4 mm are shown in Table 1. A serious problem with the empirical formula (66) that will become apparent later in this paper is that it does not include the flow velocity of the air in the duct. This leads to the contradiction that while the modal corrections in ISO 5136:199 [1] are calculated

24 4 for a duct with no flow using Eq. (66), the velocity corrections are calculated for a duct with no cross modes using Neise's version of Eq. (6). It will be shown later that this approach leads to large errors in the total correction factor at the high frequencies. It is also more satisfactory to use theoretical directivity formulae like Eqs. (3) to (6) rather than an empirical formula like Eq. (66). Indeed, the main aim of this section is to show that the theoretical Eqs. (3) to (6) are in better agreement with the experimental results than Eq. (66). The directivities of three microphone turbulence screens were measured in an anechoic room using third octave bands of random noise from 5 Hz to 1 khz. The lining and testing of this anechoic room has already been described in the journal literature [16,17]. A 3 mm diameter dual cone loudspeaker mounted in a baffle was placed 3.6 m from the centre of the microphone turbulence screen slit with the axis of the loudspeaker on the line joining the loudspeaker and the microphone turbulence screen. The loudspeaker was driven with pink noise which was passed through a third octave graphic equaliser set to boost the high and low frequency noise. The measured results were corrected for the effects of background noise. The frequency response of the microphone turbulence screens was measured from to 75 in steps of 15. The directivity relative to the response was then calculated for 15 to 75 in steps of 15. The values of the directivity expressed in decibels calculated using the different formulae were subtracted from the experimental directivity in decibels. The root mean square (rms) value of the 1 differences in decibels (4 frequencies times 5 angles) was calculated for each of the formulae. The results for three different microphone turbulence screens and the rms values across all three microphone turbulence screens (36 differences) are shown in Table. The results for a Brüel and Kjær Type UA436 microphone turbulence screen are shown in column of Table. The theoretical directivity value used in the last row was calculated using Eqs. (4) and (34). A value of 369 mks rayls was measured in situ for the specific airflow resistance of the material covering the slit. This value of specific

25 5 airflow resistance was used in the theoretical calculations. The empirical Eq. (66) and the different empirical constants shown in Table 1 were used for the other rows. Directivity measurements were also made on a microphone turbulence screen with anechoic terminations at both ends of the slit [7, 8]. The anechoic terminations consisted of 3 m of plastic tubing with the same internal cross sectional area as the slit tube. The microphone was flush mounted in the wall of the tube so that it caused no reflections. The slit was 5 mm long and 1 mm wide. The internal cross section of the slit tube was rectangular and measured 11. by 11.8 mm. Measurements were made with two different values of specific airflow resistance material covering the slit. Column 3 of Table shows the results for a specific airflow resistance extrapolated to zero flow rate of 33 mks rayls. Column 4 shows the results for 493 mks rayls. The theoretical results were calculated using Eq. (3) with a value of a equals 1 and Eq. (34). The empirical values were calculated using the first part of Eq. (66) with a slit length L of 5 mm. The values of K were calculated from the K' values using Eq. (67) with a temperature of C and L equals 4 mm. Examination of Table shows that the theoretical directivity equation agrees much better with the experimental directivity results than any of the versions of the empirical directivity equation. This implies that the theoretical directivity equation should be used instead of the empirical directivity equation when calculating the modal correction factor. When the individual directivity values were examined, none of the microphone turbulence screens completely satisfied the ISO upper and lower limiting directivity equations. Thus there is a need to revise these equations in ISO 5136 [1, ]. 6. MODAL CORRECTION FACTORS WITH NO FLOW Previously modal correction factors for ducts with no flow have been calculated using what will be referred to as the deterministic method. The modal sound pressure squared due to each propagating mode has been calculated at a point on the measurement path.

26 6 The empirical directivity formula of the microphone turbulence screen has then been applied to each modal pressure squared using the angle of incidence of the mode and the results added together to give the total sound pressure squared at the point, using the assumption that the modes are uncorrelated. The total sound pressure squared was then averaged over a number of points on the measurement path. This averaged total sound pressure squared was used to calculate the sound power being propagated down the duct by assuming that only plane wave propagation was occurring. The sound power carried down the duct by each mode was also calculated and the results added together to give the actual sound power propagating down the duct. The correction factor was given by the difference between the actual sound power and the sound power calculated from the averaged total sound pressure squared using the plane wave propagation assumption. The calculated correction factors were then averaged over a number of frequencies in each third octave band. This deterministic method does involve averaging over the circular traverse and the third octave band of frequencies. Since it is applied to a range of duct diameters, it should also involve averaging over duct diameters. It is very demanding computationally and it takes longer to complete the calculations at high frequencies in large cross sectional area ducts because the number of propagating modes becomes large. According to Holste and Neise [18], the calculated modal correction factors were extrapolated to higher frequencies and larger duct diameters for use in ISO 5136:199 [1]. The method of calculating the modal correction factor described in section 3 will be referred to in this paper as the statistical method. Its advantage is that it does not need very much computing. The results from the statistical method presented in this paper were performed in a spreadsheet. According to Neise, Frommhold, Mechel and Holste [15], the modal correction factors in ISO 5136:199 [1] were calculated using the assumption that each mode carries equal sound power down the duct. This assumption is equivalent to the assumption that the average over the cross sectional area of each modal sound pressure squared is

27 7 proportional to 1/cosθ, where θ is the angle of the direction of propagation of the mode to the centre line of the duct. If each mode has equal energy density, the average mentioned in the last sentence is constant across modes. In ISO 5136:199 [1], modal corrections are given for six different ranges of circular cross section duct diameters. The smallest three ranges in the standard use a relative measurement radius of R equals.8, whilst the largest three ranges use a relative measurement radius of R equals.65. Bolleter calculated the modal correction factors using the deterministic approach for both the upper and lower directivity limit curves given in ISO 5136:199 [1]. The modal corrections in ISO 5136:199 [1] lie on a smooth curve roughly half way between the modal correction curves for the upper and lower directivity limits. The ISO 5136:199 [1] corrections were limited to a minimum value of db and a maximum value of 6 db. In Fig., Bolleter s calculated modal corrections are compared with modal corrections calculated using the statistical method, the empirical upper and lower directivity limits of ISO 5136:199 and the assumption that each propagates equal sound power along the duct. At high frequencies, Bolleter s modal corrections are less than those of the statistical method because he limited the inverse cosine increase caused by the assumption of equal modal power to a maximum factor of 3.1. At other frequencies the agreement is reasonably good. Neise, Frommhold, Mechel and Holste [15] have calculated modal correction factors for rectangular cross section ducts with aspect ratios of 1:1, :1 and 3:1. The larger of the duct cross section dimensions took the values.5,.5, 1 and m. Calculations were made for measurement rectangles with relative radii of both.4 and.6. The empirical directivity Eq. (66) was used with an empirical constant of K' equals.5. The equal modal energy density model was assumed for the sound energy angular distribution. The calculations were made using the deterministic method described above. For comparison purposes the modal corrections were recalculated with the statistical method outlined in this paper using the same assumptions for angular sound

28 8 energy distribution and microphone turbulence screen directivity. A typical result is shown in Fig. 3 for a duct measuring.5 by.5 m. It is seen that there is fairly good agreement between the results calculated by the two different methods. However the deterministic results appear to be systematically larger than the statistical results for the high frequencies. This systematic difference was much smaller for the larger duct sizes. It is believed to be due to the fact that the cross modes first start propagating at 9 to the axis of the duct as frequency is increased. This means that the discrete individual modes propagate slightly closer to 9 than predicted by the continuous distribution of the statistical approach. Thus the deterministic model produces slightly greater values of modal correction factor than the statistical model because of the directivity of the microphone turbulence screen. As the number of cross modes increases with increasing frequency and increasing duct size this difference decreases. It should be pointed out that the results in Fig. 3 are given only for the purpose of comparing the deterministic method with the statistical method. It is believed that the assumptions used give modal correction factors which are too small. In particular it has already been shown that the empirical directivity equation does not agree as well with the experimental directivity results as the theoretical directivity equation. Fig. 4 shows the comparison of the modal corrections with no flow calculated using the statistical method, theoretical directivity and equal modal power or equal modal energy density with ISO 5136:3 and ISO 5136:199 for a circular duct of diameter.16 m. Fig. 4 can be directly compared with Fig.. Fig 4 shows that the assumption of equal modal power gives higher corrections than the assumption of equal modal energy density. 7. MODAL CORRECTION FACTORS WITH FLOW The combined modal and flow velocity correction for ISO 5136:3 [] was calculated by Arnold (see Neise and Arnold [18]). Arnold used the deterministic method with the theoretical rather than the empirical directivity model. He used the

29 9 assumption of equal modal energy density rather than equal modal power. ISO 5136:3 [] gives a combined modal and velocity correction for nose cones and foam balls of log(1 M ) db, (68) where M is the Mach number. This correction takes account of the effects of the flow on the propagation of sound power in the duct. It is added to all the theoretical calculations given in this section. In this section calculated modal correction factors with flow will be compared with experimental measurements on fans made by Holste and Neise and by Bolton. The statistical method will be used with the theoretical directivity given by Eqs. (4) and (34) and the assumption of equal modal energy density (for Holste and Neise and for Bolton) or equal modal power (for Bolton only). The in-situ experimentally measured specific airflow resistance of the material covering the slit of 369 mks rayls was used in these calculations. The experimental difference between the fan sound power determined in a free field anechoic room and the fan sound power determined in a duct are shown in Figs. 3 and 4 of Holste and Neise [19]. The numerical values for the case without cone were kindly faxed to me by Neise. In this case the values corrected for background noise were used. The values for the case with a cone attached to the fan for the anechoic room measurements were read from Fig. 3 of Holste and Neise [19]. Because the in-duct measurements used in calculating the results would have already had the modal corrections and the flow velocity corrections given by ISO 5136:199 [1] added to them, these differences are equal to the differences between the experimentally determined correction factors and the ISO 5136:199 [1] correction factors. In other words, the ISO 5136:199 [1] correction factors correspond to zero on these graphs. For this reason the modal corrections and flow velocity corrections of ISO 5136:199

30 3 [1] were subtracted from the statistical modal corrections with flow before comparing them with the experimental differences. The in-duct measurements were made in a duct with an inner diameter of 5 mm. The cut on frequency for the first cross mode is 4 Hz. Since, in this paper we are not interested in the radiation efficiency of the duct end, the results are presented only from 4 Hz upwards. The results are shown in Figs. 5(a)-(e). Figs 5(a)-(d) correspond to Figs. 3(a)-(d) of Holste and Neise [19] respectively. Both inlet and outlet measurements are shown. The theoretical results and the experimental results with and without cone show that there is a big difference between the corrections for outlet ducts and inlet ducts, since the ISO 5136:199 [1] corrections, which correspond to zero in graphs, are almost the same for outlet and inlet ducts. In ISO 5136:199 [1], most of the small differences that occur are in the wrong direction with the inlet corrections being larger than the outlet corrections. The reason for this error is that Eqs. (4) and (34) show that it is not possible to separate the modal corrections and the velocity corrections as has been done in ISO 5136:199 [1]. The effect of the flow in the outlet duct is to decrease the wave number component parallel to the axis of the duct [see Eq. (34)]. This makes the turbulence screen more directional and the correction factor larger for a given measurement frequency. In the inlet duct, the flow increases the component of the wave number which is parallel to the axis of the inlet duct. This makes the turbulence screen less directional and the correction factor smaller for a given measurement frequency. Apart from the axial flow fan in Fig. 5(d) (which also produced the highest linear flow velocity in the measurement duct ), the experimental results are generally significantly greater than the theoretical results. Nevertheless, the theoretical results are in the right direction and substantially reduce the discrepancy especially for outlet ducts at high frequencies. The theoretical and experimental inlet duct values are closer to the ISO 5136:199 [1] values (zero on these figures) than the outlet duct values

31 31 Bolton [] made measurements in a 61 mm diameter inlet duct with a Brüel and Kjær microphone turbulence screen type UA 436 and a Brüel and Kjær 1.7 mm microphone nose cone type UA 386 at a relative radius of R equals.65, and with a Brüel and Kjær polyurethane foam ball windscreen type UA 37 and a Brüel and Kjær 1.7 mm microphone nose cone type UA 386 at a relative radius of R equals.5. The linear flow velocity in the duct was 13.7 m/s. The difference in relative radius was ignored since all three non microphone turbulence screen measurements were similar. The three non microphone turbulence screen measurements were averaged and the microphone turbulence screen measurements were subtracted from this average to give an experimental estimate of the modal correction factor with flow. The theoretical results are calculated with the statistical method using the theoretical directivity formulae (4) and (34) and the assumption of equal sound power propagation down the duct by each mode or equal modal energy density. The cut on frequency for the first cross mode is 33 Hz. The experimental and theoretical corrections are shown in Fig. 6 together with the combined corrections from ISO 5136:199 [1] and ISO 5136:3 []. In this inlet duct the theory and both standards produce fairly similar corrections. (This would not be the case in the outlet duct at this flow rate as is shown in Fig. 7.) The trend of these corrections is the same as the experimental results, but they significantly underestimate the experimental results in the 1.5 to 4 khz range and overestimate the experimental results above 5 khz. This underestimation and overestimation is believed to be due to the actual angular distribution of the incident sound power on the microphone turbulence screen being different from that assumed in the theoretical models. The modal and flow velocity corrections given in Table D.1 of ISO 5136:3 [] were compared with those calculated using the statistical method developed in this paper. The assumptions were the same as those used by Arnold. In particular the specific airflow resistance of the material covering the slit was assumed to be equal to the characteristic impedance of air and the modes were assumed to have equal energy